Contents

Idea

In a strict sense of the term, a function is a homomorphismf:S→Tf : S \to T of sets. We may also speak of a map or mapping, but those terms are used in other ways in other contexts.

A function from a setAA to a set BB is determined by giving, for each element of AA, a specified element of BB. The process of passing from elements of AA to elements of BB is called function application. The set AA is called the domain of ff, and BB is called its codomain.

A function is sometimes called a total function to distinguish it from a partial function.

More generally, every morphism between objects in a category may be thought of as a function in a generalized sense. This generalized use of the word is wide spread (and justified) in type theory, where for SS and TT two types, there is a function type denoted S→TS \to T and then the expression f:S→Tf : S \to T means that ff is a term of function type, hence is a function.

Foundations

The formal definition of a function depends on the foundations chosen.

In material set theory, a function ff is often defined to be a set of ordered pairs such that for every xx, there is at most one yy such that (x,y)∈f(x,y)\in f. The domain of ff is then the set of all xx for which there exists some such yy. This definition is not entirely satisfactory since it does not determine the codomain (since not every element of the codomain may be in the image); thus to be completely precise it is better to define a function to be an ordered triple (f,A,B)(f,A,B) where AA is the domain and BB the codomain.

In structural set theory, the role of functions depends on the particular axiomatization chosen. In ETCS, functions are among the undefined things, whereas in SEAR, functions are defined to be particular relations (which in turn are undefined things).

For classes

One can also speak of functions between proper classes, although the precise details may vary depending on the status of classes with respect to the formal theory.

In ZFC for example, proper classes are by design not formal objects in the theory; rather they are proxied by a formula in the language (for instance, the class VV of all sets is proxied by the formula x=xx = x; intuitively we may think of VV as consisting of all xx satisfying that formula). Then functions f:A→Bf: A \to B between classes are again classes given by suitable formulas; see for example the MathOverflow discussion what-are-maps-between-proper-classes. If (as described above for material set theory) one wants to describe a function as an ordered triple (f,A,B)(f, A, B), then this too can be accommodated if one defines ordered triples/pairs of classes appropriately; see here for one possibility. Thus functors between categories whose objects and morphisms form proper classes can similarly be described in the language.

Such technical hacks can be avoided by choosing a different foundations. For example, Mac Lane in his Categories for the Working Mathematician assumes ZFC with a universe in which some sets are considered large, such as the set of small sets, so that a category like SetSet (the category of small sets) is again a formal object of the theory.

Alternative terms

Useful terms, more or less synonymous with function, are assignment, assignation or more specifically assignation on objects. These do not have standard meanings but are useful to signal to readers that the domain of the ‘function’ under consideration is large, or that one is more interested in functorial extensions of this partial assignation (cf. e.g. Richard Garner, Homomorphisms of higher categories, Adv. Math. 224 (2010) 2269-2311 for many examples). In mathematical writing “assignment” is usually synonymous with function or map or “mapping”. For example one might speak of “assigning to each positive number its square root” to refer to the function (−):ℝ≥0→ℝ\sqrt{(-)} \colon \mathbb{R}_{\geq 0} \to \mathbb{R}.

Authors may resort to verb forms such as “assigns” or “associates” or “sends” in informal writing, perhaps to avoid the bother of specifying an axiomatic framework in which a formal notion like “function” is ensconced. For example, according to Wikipedia, Jacobson defines a functor F:C→DF: C \to D between categories as a “mapping” that “associates” to each object XX in CC an object F(X)F(X) in DD, etc. No clarity would be gained by making this any more formal (which as we saw in the case of functions between classes, such as classes of objects of categories, may involve annoyingly technical hacks).

Sometimes the word “assignment” is understood more generally as relation, often when authors define a function to be something that “assigns unique values” (for instance here).